Zigzag instability and axisymmetric rolls in Rayleigh-Bénard convection: The effects of curvature

Abstract

Using a formalism adapted to slow spatio-temporal evolutions of phase perturbations to a perfect roll pattern, we show that zigzags which relax in a diffusive way for the case with (or rigid) boundaries are replaced by Busse's oscillatory mode for the case with boundaries. The diffusion coefficient ${D}_{\ensuremath{\perp}}$ for zigzags, which is determined analytically using a perturbation expansion, has the form ${D}_{\ensuremath{\perp}}=\frac{{\ensuremath{\xi}}_{0}^{2}}{{\ensuremath{\tau}}_{0}}\left[\frac{q\ensuremath{-}{q}_{c}}{{q}_{c}}+\frac{N(\mathcal{P})}{{R}_{2}(\mathcal{P})}\frac{\mathcal{R}\ensuremath{-}{\mathcal{R}}^{c}}{{\mathcal{R}}^{c}}\right],$ where $\mathcal{R} [={\mathcal{R}}^{c}+{R}_{2}(\mathcal{P}){\ensuremath{\epsilon}}^{2}+\ensuremath{\cdots}]$ is the Rayleigh number, $\mathcal{P}$ is the Prandtl number, ${\ensuremath{\xi}}_{0}$ is the coherence length, and ${\ensuremath{\tau}}_{0}$ is the relaxation time; $\frac{{\ensuremath{\xi}}_{0}^{2}}{{\ensuremath{\tau}}_{0}}=\frac{5.685}{(1.95+{\mathcal{P}}^{\ensuremath{-}1})}$, $N(\mathcal{P})=0.166+23.04{\mathcal{P}}^{\ensuremath{-}1}+6.196{\mathcal{P}}^{\ensuremath{-}2}$, ${R}_{2}(\mathcal{P})=10.76\ensuremath{-}0.073{\mathcal{P}}^{\ensuremath{-}1}+0.128{\mathcal{P}}^{\ensuremath{-}2}$, and ${\mathcal{R}}^{c}=1707.8$ at the critical wave vector ${q}_{c}=3.116$ for no-slip boundaries. The curvature is seen to induce a large-scale flow which tends to convect the roll pattern. This flow is forbidden for concentric rolls in cylindrical geometry. This leads to a new wavelength-selection criterion which no longer corresponds to ${D}_{\ensuremath{\perp}}=0$. The corrected criterion is calculated for both no-slip and stress-free boundaries and is seen to behave in a similar way at the limit $\mathcal{P}\ensuremath{\rightarrow}0$ in the two cases. Our findings are in quantitative agreement with earlier numerical results and also with more recent approaches using the amplitude-equation formalism.